Self-dual Functions Research Articles

Structure of clones in the precomplete class of self-dual functions in three-valued logic

Structure of clones in the precomplete class of self-dual functions in three-valued logic

Mixed id-expansion of multivalued logic functions

The conception of id-expansion of multivalued logic function is introduced. For any k ≥ 2, the degree of mixed id-expansion of the class of all k-valued logic functions and the degree of mixed id-expansion of the class of all Boolean functions over the class of self-dual monotonic functions are defined.

Invariant continuous aggregation functions

Continuous aggregation functions, being invariant under any monotone bijective transformation, are discussed and characterised. As a basic tool for this characterisation, self-dual -valued capacities or their counterparts, self-dual monotone Boolean functions, are exploited. A generating role of is discussed. The relevant role of simple medians in describing monotone threshold Boolean functions is made apparent.

Hamiltonian systems of PDEs with selfdual boundary conditions

Selfdual variational calculus is developed further and used to address questions of existence of local and global solutions for various parabolic semi-linear equations, and Hamiltonian systems of PDEs. This allows for the resolution of such equations under general time boundary conditions which include the more traditional ones such as initial value problems, periodic and anti-periodic orbits, but also yield new ones such as “periodic orbits up to an isometry” for evolution equations that may not have periodic solutions. In the process, we introduce a method for perturbing selfdual functionals in order to induce coercivity and compactness, without destroying the selfdual character of the system.

Dichotomy Results for Fixed-Point Existence Problems for Boolean Dynamical Systems

A complete classification of the computational complexity of the fixed-point existence problem for Boolean dynamical systems, i.e., finite discrete dynamical systems over the domain {0, 1}, is presented. For function classes $${\mathcal{F}}$$ and graph classes $${\mathcal{G}}$$ , an ( $${\mathcal{F}},{\mathcal{G}}$$ )-system is a Boolean dynamical system such that all local transition functions lie in $${\mathcal{F}}$$ and the underlying graph lies in $${\mathcal{G}}$$ . Let $${\mathcal{F}}$$ be a class of Boolean functions which is closed under composition and let $${\mathcal{G}}$$ be a class of graphs which is closed under taking minors. The following dichotomy theorems are shown: (1) If $${\mathcal{F}}$$ contains the self-dual functions and $${\mathcal{G}}$$ contains the planar graphs, then the fixed-point existence problem for ( $${\mathcal{F}},{\mathcal{G}}$$ )-systems with local transition function given by truth-tables is NP-complete; otherwise, it is decidable in polynomial time. (2) If $${\mathcal{F}}$$ contains the self-dual functions and $${\mathcal{G}}$$ contains the graphs having vertex covers of size one, then the fixed-point existence problem for ( $${\mathcal{F}},{\mathcal{G}}$$ )-systems with local transition function given by formulas or circuits is NP-complete; otherwise, it is decidable in polynomial time.

Lee weight enumerators of self-dual codes and theta functions

The theory of modular forms, in particular theta functions, and coding theory are in a remarkable way connected. The connection is established by defining a suitable lattice corresponding to the given code, and considering its theta function. First we define some special theta functions, and determine transformation formulas. Then it is proved that the theta function of the lattice corresponding to the code can be expressed in terms of the Lee weight enumerator. In particular if the code is self-dual and the length is a multiple of $8$ this theta function is a modular form for some subgroup of the modular group. Using the known structure of this space of modular forms we can derive linear relations between the coefficients of the Lee weight enumerator. And from these relations we can get an upper bound for the minimal Lee distance of self-dual $p$-ary codes.

Superposition of selfdual functionals in non-homogeneous boundary value problems and differential systems

Selfdual variational theory -- developed in [8] and [9] -- allows for the superposition of appropriate 'boundary' Lagrangians with 'interior' Lagrangians, leading to a variational formulation and resolution of problems with various linear and nonlinear boundary constraints that are not amenable to standard Euler-Lagrange theory. The superposition of several selfdual Lagrangians is also possible in many natural settings, leading to a variational resolution of certain differential systems. These results are applied to nonlinear transport equations with prescribed exit values, Lagrangian intersections of convex-concave Hamiltonian systems, initial-value problems of dissipative systems, as well as evolution equations with periodic and anti-periodic solutions.

Self-Dual Self-Testing Multicycle Circuits: Their Properties

Functional testing of memory circuits based on the properties of self-dual functions and a procedure for transforming the initial circuit into a self-dual circuit are described. Experimental results for MCNC benchmark circuits are given.

A polynomial algorithm for the recognition of belonging a function of k-valued logic realized by a polynomial to precomplete classes of self-dual functions

A polynomial algorithm for the recognition of belonging a function of k-valued logic realized by a polynomial to precomplete classes of self-dual functions

A Parametric Approach to Efficiency Measurement Using a Flexible Profit Function

Although the idea of measuring productive efficiency goes back to Farrell [8], the econometric estimation of it began with Aigner, Lovell, and Schmidt [1] and Meeusen and van den Broeck [17]. In Aigner, Lovell, and Schmidt [1] and Meeusen and van den Broeck [17], and their extensions which incorporate either cost minimizing or profit maximizing behavior, efficiency is measured relative to a frontier. An alternative to the frontier approach that started with Hopper [10] and Lau and Yotopoulos [15], is to measure efficiency (mostly allocative) without estimating any frontier. In the latter approach, allocative inefficiency (defined as the deviations of the first order conditions of profit maximization or cost minimization) is modeled through shadow (virtual) prices which are parametric functions of observed prices. The shadow price approach is extensively used in the literature on efficiency measurement because it has an advantage over the frontier approach, which requires distributional assumptions on the error terms-especially in cross sectional models. In empirical studies on the efficiency of regulated utilities, it has been argued that due to the presence of rate of return and other regulations, these firms optimize with respect to shadow prices instead of observed prices. These studies often fail to consider the possibility that the firms under question may be technically inefficient as well, which can affect allocation of inputs.1 This issue is important in both cost minimization and profit maximization cases. This paper uses a profit maximization framework and develops a generalized profit function approach that accommodates both technical and allocative inefficiencies in the context of a panel data model. The relationship between production technical inefficiency (loss of output due to technical inefficiency) and profit technical inefficiency (profit loss due to technical inefficiency) is derived for a flexible production function. Presently, it is believed that the derivation of this relationship is only possible for the self-dual production functions. We show that if the profit function is translog, some popular approaches used in modeling technical and allocative inefficiency are incorrect. In particular, we show that: (i) models that fail to include technical inefficiency and consider only allocative inefficiency yield biased and inconsistent parameter estimates; (ii) if technical inefficiency is neglected, the measure of technical change will be biased. The generalized profit function developed here is capable of distinguishing between technical change and time-varying technical inefficiency. Such a distinction is not possible if, for example, one uses

Linear Separability of Positive Self-Dual Logical Filters

The threshold logic filter is an important nonlinear filter class, which is defined by a threshold logic function of binary input values. It is proved in this paper that not all positive self-dual logical functions are threshold functions if the number of input variables is 5 or more. The positive self-dual logical filter is a limited class but includes almost all filters for noise removal. Our result means that not all positive self-dual filters can be expressed by one operation of the threshold logic filters. In the sense of the filter expression by neural networks, the two-layer network cannot always optimize even this limited class.

Decompositions of positive self-dual boolean functions

A coterie, which is used to realize mutual exclusion in distributed systems, is a family C of subsets such that any pair of subsets in C has at least one element in common, and such that no subset in C contains any other subset in C. Associate with a family of subsets C a positive Boolean function f c such that f c ( x) = 1 if the Boolean vector x is equal to or greater than the characteristic vector of some subset in C, and 0 otherwise. It is known that C is a coterie if and only if f c is dual-minor, and is a non-dominated (ND) coterie if and only if f c is self-dual. We study in this paper the decomposition of a positive self-dual function into smaller positive self-dual functions, as it explains how to represent and how to construct the corresponding ND coterie. A key step is how to decompose a positive dual-minor function f into a conjunction of positive self-dual functions f 1, f 2,…, f k. In addition to the general condition for this decomposition, we clarify the condition for the decomposition into two functions f 1, and f 2, and introduce the concept of canonical decomposition. Then we present an algorithm that determines a minimal canonical decomposition, and a very simple algorithm that usually gives a decomposition close to minimal. The decomposition of a general self-dual function is also discussed.

Generating and approximating nondominated coteries

A coterie, which is used to realize mutual exclusion in a distributed system is a family C of incomparable subsets such that every pair of subsets in C has at least one element in common. Associate with a family of subsets C a positive (i.e., monotone) Boolean function f/sub c/ such that f/sub c/(x)=1 if the Boolean vector x is equal to or greater than the characteristic vector of some subset in C, and 0 otherwise. It is known that C is a coterie if and only if f/sub c/ is dual-minor, and is a nondominated (ND) coterie if and only if f/sub c/ is self-dual. We introduce an operator /spl rho/, which transforms a positive self-dual function into another positive self-dual function, and the concept of almost-self-duality, which is a close approximation to self-duality and can be checked in polynomial time (the complexity of checking positive self-duality is currently unknown). After proving several interesting properties of them, we propose a simple algorithm to check whether a given positive function is self-dual or not. Although this is not a polynomial algorithm, it is practically efficient in most cases. Finally, we present an incrementally polynomial algorithm that generates all positive self-dual functions (ND coteries) by repeatedly applying p operations. Based on this algorithm, all ND coteries of up to seven variables are computed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Agricultural Productivity in Africa: Discussion

can (SSA) agriculture. Each adds to our understanding of processes of change in SSA agricultural production, cost, and productivity, although major gaps in that understanding remain. While each paper relies on different data and methods, taken together they stand in some contrast to most studies of SSA agriculture over the last decade in that they each offer more positive readings of change in SSA agriculture than have most previous studies. I will discuss each study briefly, beginning with the micro-data study (Savadogo, Reardon, and Pietola for Burkina Faso), then the macrodata study of Zimbabwe (Jayne, et al.), and finally Black's multi-country data study (hereafter referred to as the Burkina, Zimbabwe, and SSA studies). The Burkina Faso paper is a first effort at analyzing a rich data set collected with considerably more care than is the norm for SSA agricultural studies. The data set in question includes crop-specific inputs (land, labor, fertilizer and manure, and, presumably, animal labor) and enables estimation of crop-specific production functions. The authors specify a quadratic production function (1) and input demand functions (2). It is not clear what is gained by the use of the complex quadratic production function, since factor demand functions consistent with it are not estimated. (The quadratic production function is not a self-dual function, although the normalized quadratic profit function is widely used as in the Zimbabwe paper.) Savadogo, Reardon, and Pietola could have treated their production functions as having an animal traction input and, in a pooled sample, attempted to deal with the problem of corner solutions, in that roughly half the farmers do not actually use animal traction. The Mills ratio from (3) would have been appropriate. Instead they chose to split the sample, creating a selectivity bias. This problem is recognized and the Mills ratio procedure is probably also appropriate for it. However, in the animal traction sample, the animal traction input is a left-out variable and this creates a bias in the estimates.

A theory of coteries: mutual exclusion in distributed systems

A coterie under a ground set U consists of subsets (called quorums) of U such that any pair of quorums intersect with each other. Nondominated (ND) coteries are of particular interest, since they are optimal in some sense. By assigning a Boolean variable to each element in U, a family of subsets of U is represented by a Boolean function of these variables. The authors characterize the ND coteries as exactly those families which can be represented by positive, self-dual functions. In this Boolean framework, it is proved that any function representing an ND coterie can be decomposed into copies of the three-majority function, and this decomposition is representable as a binary tree. It is also shown that the class of ND coteries proposed by D. Agrawal and A. El Abbadi (1989) is related to a special case of the above binary decomposition, and that the composition proposed by M.L. Neilsen and M. Mizuno (1992) is closely related to the classical Ashenhurst decomposition of Boolean functions. A number of other results are also obtained. The compactness of the proofs of most of these results indicates the suitability of Boolean algebra for the analysis of coteries.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Analysis of weighted median filters based on inequalities relating the weights

By considering all orderings of the input samples, which are discrete-time continuous-valued, it is shown here that a weighted median (WM) filter of spanN can be specified unambiguously by 2N−1 consistent linear inequalities relating the weights. This specification is identical to that of a self-dual threshold function with the same weights. It is also shown that WM filters with symmetric weights can be specified by ternary threshold functions. Based on these inequalities, properties of WM filters which can be used to check equivalence of some WM filters are derived.

Self-Dual Polynomials, Hermite Matrices, and Heisenberg Functionals

For certain orthogonal matrices associated with classical self dual discrete orthogonal families, commuting symmetric tridiagonal matrices are constructed. Their eigenvectors are shown to be critical points for functionals related to Heisenberg’s inequality.

Passive network synthesis via dual spectral factorization

The role of dual transfer function matrices in network analysis and synthesis is described. The transfer function matrices are actually spectral factors of a certain power spectrum matrix. Procedures for constructing dual and self-dual transfer function matrices are presented, from which general synthesis methods are derived for nonreciprocal and reciprocal networks. A number of known synthesis procedures are shown to be special instances of the general methods.

A Note on Minimal Reed-Muller Canonical Forms of Switching Functions

A nonexhaustive procedure for obtaining minimal Reed-Muller canonical (RMC) forms of switching functions is presented. This procedure is a modification of a procedure presented earlier in the literature and enables derivation of an upper bound on the number of RMC forms to be derived to choose a minimal one. It is shown that the task of obtaining minimal RMC forms is simplified in the case of symmetric functions and self-dual functions.

Boundary Points of Threshold Functions

Boundary points of a threshold function fare those vertices of the n-cube that produce a minimal irredundant set of inequalities for the realization of f. In this paper they are shown to be those vertices that can be contained in separating hyperplanes. With this theorem it is shown that knowledge of canonical boundary points of self-dual canonical threshold functions allows complete determination of boundary points of equivalent functions. This provides a compact and geometrically interesting characterization of threshold functions.