Shannon Functions Research Articles

Bounds on Shannon functions of lengths of contact closure tests for contact circuits

Abstract We consider the problem of synthesis of irredundant two-pole contact circuits which implement n-place Boolean functions and allow short single fault detection or diagnostic tests of closures of at most k contacts. We prove that the Shannon function of the length of a fault detection test is equal to n for any n and k, and that the Shannon function of the length of a diagnostic test is majorized by n + k(n − 2) for n ⩾ 2.

Tests with Stuck-At and Shift Faults on Circuit Inputs

We investigate left-shifts of variables in Boolean functions and stuck-at faults. Some bounds of Shannon functions are established, producing the minimum test lengths for detection and diagnosis of any n-variable Boolean function with respect to a given fault source. (The Shannon function of the test length is the maximum of the minimum test lengths over all n-variable Boolean functions.) A left-shift in an n-variable Boolean function simultaneously increments the indexes of all the variables by a natural number and substitutes some constants for variables with indexes greater than n. The Shannon function of the relevant test is shown to be exactly of length 2; the order of growth of the Shannon function of the length of the diagnosis test is $$ \Theta \left(\sqrt{2^n}\right). $$ With the variables left-shifted by k places, 1 ≤ k ≤ n, the Shannon function $$ {L}^{diagn}\left({U}_{n,k}^{shifts},n\right) $$ of the length of the complete diagnosis test has the following bounds: $$ \min \left({2}^k-1,{2}^{n-k}\right)\le {L}^{diagn}\left({U}_{n,k}^{shifts},n\right)\le \min \left({2}^k,{2}^{n-k}+1\right). $$ The article also considers local stuck-at faults of multiplicity k on the inputs of k-input, one-output circuits realizing a Boolean function. An asymptotic expression is derived for the log Shannon function of the diagnosis test length $$ {\log}_2{L}^{diagn}\left({U}_k^{lc},n\right)\sim k\ \mathrm{as}\ n\to \infty, k=k(n)\to \infty, \kern1em 1\le k\le n/2,{\log}_2n=o(k). $$ An asymptotic lower bound of the Shannon function of the diagnosis test length with stuck-at faults on circuit inputs is shown to be Ldiagn(Uc, n) ≥ 2 ⋅ 2[n/2] ⋅ (1 + o(1)) as n → ∞ .

Climate variability in early expansions of Homo sapiens in light of the new record of micromammals in Misliya Cave, Israel

In this study, we provide the first taphonomic and taxonomic descriptions of the micromammals from Misliya Cave, where recently a Homo sapiens hemimaxilla has been reported. This finding significantly extends the time frame for the out-of-Africa presence of anatomically modern humans. It also provides an opportunity to reassess variation in early modern human population responses to climate change in the Levantine sequence. Information on species ranking and diversity estimations (Shannon functions) is obtained from quantitative data across 31 Levantine assemblages and investigated in a broad comparative frame using multivariate analyses. Recent models of human-climate interactions in the late Early–Middle Paleolithic of the southern Levant have drawn heavily on on-site associations of human fossils with remains of micromammals. However, there has been little, if any, attempt to examine the long-term picture of how paleocommunities of micromammals responded qualitatively and quantitatively to climatic oscillations of the region by altering their compositional complexity. Consequently, our understanding is vastly limited in regard to the paleoecosystem functions that linked past precipitation shifts to changes in primary producers and consumers or as to the background climatic conditions that allowed for the development of highly nonanalog ancient communities in the region. Although previous studies argued for a correspondence between alternations in H. sapiens and Neanderthal occupations of the Levant and faunal shifts in key biostratigraphic indicator taxa (such as Euro-Siberian Ellobius versus Saharo-Arabian Mastomys and Arvicanthis), our data indicate the likelihood that early H. sapiens populations (Misliya and Qafzeh hominins) persisted through high amplitudes of paleoecological and climatic oscillations. It is unlikely, given these results, that climate functioned as a significant filter of early modern human persistence and genetic interactions with Neanderthals in the Levant.

The complexity of Boolean functions in the Reed–Muller polynomials class

This paper considers the problem of transforametion of Boolean functions into canonical polarized polynomials (Reed–Muller polynomials). Two Shannon functions are introduced to estimate the complexity of Boolean functions in the polynomials class under consideration. We propose three Boolean functions of n variables whose complexity (in terms of the number of terms) coincides with value. We investigate the properties of functions and propose their schematic realization on elements AND, XOR, and NAND.

Global and Regional Gravity Field Determination from GOCE Kinematic Orbit by Means of Spherical Radial Basis Functions

We present global and regional gravity field models to degree 130 based on the GOCE kinematic orbit from the period 01 November 2009 to 11 January 2010. The gravity field models are parameterized in terms of the Shannon and Kaula's spherical radial basis functions. The relation between the unknown expansion coefficients and the kinematic orbit of the satellite is established by the acceleration approach. We show that our global GOCE-only solutions free from prior information can compete with unconstrained spherical harmonic models in terms of accuracy. Furthermore, we utilize our low-degree global GOCE-based models to introduce prior information into the least-squares adjustment. This procedure substantially improves the zonal and near-zonal spherical harmonic coefficients, which are usually degraded due to the polar gap problem. As an unwanted side effect, low-pass filtering of the geopotential may occur, but this can be adjusted by the spectral content of the prior information. We show that the regional enhancement of the global solutions reduces noise in the final model between degrees 70 and 130 by ~10 % in terms of RMS error. In general, our Shannon-based solutions systematically outperform the Kaula-based ones. To validate our results, we use the EIGEN-6S model, which is superior to the solutions from kinematic orbits at least by one order of magnitude. Both the global and the regional models satisfy the GOCE-only strategy.

Orders of growth of Shannon functions for circuit complexity over infinite bases

It is shown that any function of one real variable being a composition of rational functions with real coefficientials, logarithms, and exponents and having an order of growth between n and \(2^{O(n^{1/2} )}\) is an order of growth of the Shannon function for the circuit complexity over a certain infinite basis.

Complexity of functions from some classes of three-valued logic

The problem of the realization complexity for functions of the three-valued logic taking values from the set {0, 1} by formulas over incomplete generating systems is considered. Upper and lower asymptotic estimates for the corresponding Shannon functions are obtained.

Depth of α-completions of systems of Boolean functions

A problem of realization of Boolean functions by α-formulas is considered. These formulas are such that each subformula contains not more than one nontrivial principal subformula. The depth is considered as a complexity measure of a formula. Upper and lower polynomial estimates of Shannon functions for α-completions of finite systems of Boolean functions are obtained.

On the complexity of representation of k-valued functions by generalised polarised polynomials

We consider generalised polarised polynomials for k-valued functions (for prime k). It is proved that each k-valued function is represented by some unique generalised polarised polynomial for each polarisation vector. We find upper and lower bounds for the Shannon functions of degree and length of the generalised polarised polynomials of k-valued functions.

Synthesis of formulas whose complexity and depth do not exceed the asymptotically best estimates of high degree of accuracy

We study problems concerning optimal realizations of arbitrary Boolean functions by formulas in the standard basis {&, V, ¬} in the presence of two optimality criteria: the depth and the complexity. Both the complexity and depth of Boolean functions are investigated from the point of view of so-called asymptotically best estimates of high degree of accuracy for the corresponding Shannon functions. Such estimates produce an asymptotics not only for the Shannon function, but also for the first remainder term of its standard asymptotic expansion. For any Boolean function depending on n variables, we prove that it is possible to construct a realizing formula in the basis {&, V, ¬} so that its depth and complexity do not exceed values of the corresponding Shannon functions for the argument equal to n in the sense of asymptotic estimates of high degree of accuracy.

On estimates of Shannon functions of the length of unit tests for transpositions of variables

A nontrival upper estimate of the form n √2n (1 + o(1)) is proposed for the Shannon function of the length of the unit checking test for transpositions of variables of a Boolean function from P 2 n , and it is proved that the Shannon function of the length of a unit diagnostic test for transpositions of variables of a Boolean function from P 2 n has an asymptotics of the form n 2/2.

Recognizing the nonrepeating property in an arbitrary basis

Two classical problems are considered: recognizing the properties of a Boolean function given a column of its values and constructing a diagnostic test. The problems are investigated for nonrepeating functions in an arbitrary basis B. For the first problem, the decomposition method is applied to prove linear complexity of the corresponding sequential circuits; for the second problem we derive the order of the Shannon functions for a number of bases, in particular, for the basis of all functions of four variables.

Average time of computing Boolean operators

The average time of computing Boolean operators by straight-line programs of two types is studied. Upper and lower bounds for the corresponding Shannon functions are obtained. Asymptotically exact formulas for Shannon functions are derived in the case when the number of components of the operators to be computed increases with the number of their arguments.

On complexity of representation of multiple-valued logic functions as polarised polynomials

AbstractThe notion of a polarised polynomial form is extended to the case of multiple-valued logic functions. We introduce the Shannon functions of weight and length of polarised polynomial forms of multiple-valued logic functions and give some bounds for them.

Light scattering by “soft” parallelepipeds of arbitrary size

A generalisation of the earlier proposed method of light scattering analysis based on the expansion into a spatial spectrum with the aid of the sampling Kotel'nikov–Shannon functions is done. This method has been applied so far only to the case of small particles in the frame of Rayleigh–Gans approximation. Being generalised, the method could be applied to the analysis of scattering from optically “soft” particles of arbitrary dimension and shape in the approach equivalent to that of WKB. The case of scattering from a parallelepiped is analysed by this method. The indicatrices and the integral cross-sections are obtained and their features discussed.

Entropia, informação e estresse hídrico em Eucaliptus camadulensis in vitro

O presente trabalho objetivou explorar uma nova linha de estudo sobre desenvolvimento de plantas por meio da interpretação de seus estados entrópicos. Em Eucalyptus camadulensis cultivados in vitro em diferentes condições de deficiência hídrica, estes foram avaliados por meio de sua produção de matéria seca utilizando-se as funções de entropia de Clausius e Shannon. O material submetido à deficiência hídrica obedeceu ao segundo princípio da termodinâmica, comportando-se como uma estrutura dissipativa. À medida que houve um aumento na produção de entropia, houve uma menor diferenciação dos tecidos foliares, ou seja, uma redução do nível de organização da planta, principalmente a partir do potencial osmótico de -1,5 MPa. A eficiência termodinâmica ideal para o material analisado encontra-se entre 0,57% e 0,62%.

Unimprovable Upper Bounds on Time Complexity of Decision Trees

The paper is devoted to investigation of behavior of global Shannon functions which produce unimprovable upper bounds on time complexity of decision trees over arbitrary information systems.

Complexity of Boolean functions in the class of polarized polynomial forms

It is proved that values of Shannon functions in the class of polarized polynomial forms (generalized Reed-Muller forms) coincide for classes of all Boolean functions and all symmetric Boolean functions; a formula computing estimates for these functions is derived.

ABOUT THE DEPTH OF DECISION TREES COMPUTING BOOLEAN FUNCTIONS

Lower and upper bounds for the depth of decision trees computing Boolean functions are established and Shannon functions of the decision tree depth are determined for closed classes of Boolean functions.

Retrieval performance and information theory

This paper challenges the meaningfulness of precision and recall values as a measure of performance of a retrieval system. Instead, it advocates the use of a normalised form of Shannon's functions (entropy and mutual information). Shannon's four axioms are replaced by an equivalent set of five axioms which are more readily shown to be pertinent to document retrieval. The applicability of these axioms and the conceptual and operational advantages of Shannon's functions are the central points of the work. The applicability of the results to any automatic classification is also outlined.