On algorithm complexity

This paper contains a review of the authors’ results in the theory of algorithm complexity. The results described concern methods for obtaining lower bounds (containing almost all exponential lower bounds on monotone complexity of monotone functions), synthesis of asymptotically optimal functional networks, minimization of Boolean functions, and the problem of solving Boolean equations.     

On the complexity of growth of the number of distinct fuzzy switching functions

Several attempts have been made to enumerate fuzzy switching (FSF's) and to develop upper and lower bounds for the number of FSF's of n variables in an effort to better understand the properties and the complexity of FSF's. Previous upper bounds are 2⁴ⁿ [9] and 2^{2–3n—2n—1} [7].It has also been shown that the exact numbers of FSF's of n variables for n = 0, 1, 2, 3, and 4 are 2, 6, 8, 84, 43 918 and 160 297 985 276 respectively.This paper will give a brief overview of previous approaches to the problem, study some of the properties of fuzzy switching functions and give improved upper and lower bounds for a general n.     

On the fluctuation spectrum of plasma

The spectrum of electron phase space density fluctuations of a plasma is calculated by a novel method that parallels conventional calculations of the partition function in statistical physics. Expressions for the electric field fluctuations and the closely related form factor agree with existing results. The method clears up ambiguities about equipartition and provides a new expression for the spectrum of electrostatic phase space density fluctuations about stable non-Maxwellian equilibria.     

On complexity of round transformations

A modern block cipher consists of round transformations, which are obtained by alternatively applying permutations (P-boxes) and substitutions (S-boxes). Clearly, the most important attribute of a block cipher is its security. However, with respect to the hardware implementation, a good block cipher has to have a reasonable complexity as well. In this paper, we study complexity of round transformations satisfying some basic security criteria. There are several ways to define the complexity of a round transformation, and to choose “necessary” security criteria. It turns out, that for our purpose, it is suitable to view a round transformation as a single Boolean function, not separating it into S-boxes and P-boxes. We require that the Boolean function F possesses some fundamental properties imposed on each block cipher for security reasons; namely, we require that the function is a strictly non-linear bijection and that it has a good diffusion. The total number of variables in the normal algebraic form of the component functions of F is taken as its complexity. We find the minimum complexity of such functions, and this way we establish a lower bound on complexity of all round transformations. To show that the lower bound is the best possible, we construct a round transformation F^′ attaining the bound. We stress that it is not an aspiration of this paper to construct a round transformation which would be of practical use; F^′ is useful only from the theoretical point of view.     

On the complexity of posets

The purpose of this paper is to discuss several invariants each of which provides a measure of the intuitive notion of complexity for a finite partially ordered set. For a poset X the invariants discussed include cardinality, width, length, breadth, dimension, weak dimension, interval dimension and semiorder dimension denoted respectively X, W(X), L(X), B(X), dim(X). Wdim(X), Idim(X) and Sdim(X). Among these invariants the following inequalities hold. $\text{B(}\text{X}\text{)?Idim(}\text{X}\text{)?Sdim(}\text{X}\text{)?Wdim(}\text{X}\text{)?dim(}\text{X}\text{)?W(}\text{X}\text{)}$. We prove that every poset X with three of more points contains a partly with $\text{Idim}\text{(}\text{X}\text{)}\text{Idim}\text{(}\text{X}\text{) {x,v})}$. If M denotes the set of maximal elements and A an arbitrary anticham of X we show that $\text{Idim}\text{(}\text{X}\text{)?W(}\text{X-M}\text{)}$ and $\text{Idim}\text{(}\text{X}\text{)?2W(}\text{X-A}\text{)}$. We also show that there exist functions f(n,t) and (gt) such that $\text{I(}\text{X}\text{)?n}$ and $\text{Idim}\text{(}\text{X}\text{)?t}$simply $\text{dim}\text{(}\text{X}\text{)?f(n,t and Sdim(}\text{X}\text{)?t}$ implies $\text{dim}\text{(}\text{X}\text{)?g(t)}$.     

On the complexity of circulations

We present a polynomial-time algorithm for producing a feasible real-valued circulation in undirected graphs with upper and lower bounds, based on Seymour's characterization. For directed graphs, an efficient algorithm follows from a classical result by Hoffman. We show that, for mixed graphs, i.e., graphs having both directed and undirected edges, the problem is NP-complete.     

On complexity of representations of quivers

It is shown that, given a representation of a quiver over a finite field, one can check in polynomial time whether it is absolutely indecomposable.     

On the relation between complexity and uncertainty

In practical problem situations data are usually inherently unreliable. A mathematical representation of uncertainty leads to stochastic optimization problems. In this paper the complexity of stochastic combinatorial optimization problems is discussed. Surprisingly, certain stochastic versions of NP-hard determinstic combinatorial problems appear to be solvable in polynomial time.     

On the complexity of enumerating pseudo-intents

We investigate whether the pseudo-intents of a given formal context can efficiently be enumerated. We show that they cannot be enumerated in a specified lexicographic order with polynomial delay unless P=NP. Furthermore we show that if the restriction on the order of enumeration is removed, then the problem becomes at least as hard as enumerating minimal transversals of a given hypergraph. We introduce the notion of minimal pseudo-intents and show that recognizing minimal pseudo-intents is polynomial. Despite their less complicated nature, surprisingly it turns out that minimal pseudo-intents cannot be enumerated in output-polynomial time unless P=NP.     

On the computational complexity of cut-reduction

Using appropriate notation systems for proofs, cut-reduction can often be rendered feasible on these notations. Explicit bounds can be given. Developing a suitable notation system for Bounded Arithmetic, and applying these bounds, all the known results on definable functions of certain such theories can be reobtained in a uniform way.     

On the complexity of constrained VC-classes

Sauer's lemma is extended to classes H_N of binary-valued functions h on [n]={1,…,n} which have a margin less than or equal to N on all x∈[n] with h(x)=1, where the margin μ_h(x) of h at x∈[n] is defined as the largest non-negative integer a such that h is constant on the interval I_a(x)=[x-a,x+a]⊆[n]. Estimates are obtained for the cardinality of classes of binary-valued functions with a margin of at least N on a positive sample S⊆[n].     

On strong embeddability and finite decomposition complexity

The strong embeddability is a notion of metric geometry, which is an intermediate property lying between coarse embeddability and property A. In this paper, we study the permanence properties of strong embeddability for metric spaces. We show that strong embeddability is coarsely invariant and it is closed under taking subspaces, direct products, direct limits and finite unions. Furthermore, we show that a metric space is strongly embeddable if and only if it has weak finite decomposition complexity with respect to strong embeddability.     

On the complexity and analysis of auxetic systems

A challenge in creating a model for an auxetic system based on a formalism that is fully computable, is the aim of this paper. Two major levels of complexity are discussed in a way of understanding the structure and processes that define an auxetic system. The auxeticity and structural complexity are interpreted in the light of Cosserat elasticity which admits degrees of freedom not present in classical elasticity, i.e. the rotation of points in the material, and a couple per unit area or the couple stress. The Young'modulus computing for a laminated periodic system made up of alternating aluminum and an auxetic material is an example of computing complexity. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On complexity of multistage stochastic programs

In this paper we derive estimates of the sample sizes required to solve a multistage stochastic programming problem with a given accuracy by the (conditional sampling) sample average approximation method. The presented analysis is self-contained and is based on a relatively elementary, one-dimensional, Cramér's Large Deviations Theorem.