Three-valued logics, uncertainty management and rough sets

This paper is a survey of the connections between three-valued logics and rough sets from the point of view of incomplete information management. Based on the fact that many three-valued logics can be put under a unique algebraic umbrella, we show how to translate three-valued conjunctions and implications into operations on ill-known sets such as rough sets. We then show that while such translations may provide mathematically elegant algebraic settings for rough sets, the interpretability of these connectives in terms of an original set approximated via an equivalence relation is very limited, thus casting doubts on the practical relevance of truth-functional logical renderings of rough sets

Lattice paths and branched continued fractions II. Multivariate Lah polynomials and Lah symmetric functions

We introduce the generic Lah polynomials Ln,k(ϕ), which enumerate unordered forests of increasing ordered trees with a weight ϕi for each vertex with i children. We show that, if the weight sequence ϕ is Toeplitz-totally positive, then the triangular array of generic Lah polynomials is totally positive and the sequence of row-generating polynomials Ln(ϕ,y) is coefficientwise Hankel-totally positive. Upon specialization we obtain results for the Lah symmetric functions and multivariate Lah polynomials of positive and negative type. The multivariate Lah polynomials of positive type are also given by a branched continued fraction. Our proofs use mainly the method of production matrices; the production matrix is obtained by a bijection from ordered forests of increasing ordered trees to labeled partial Łukasiewicz paths. We also give a second proof of the continued fraction using the Euler–Gauss recurrence method

Translational Oncogenomics and Human Cancer Interactome Networks

An overview of translational, human oncogenomics, transcriptomics and cancer interactomic networks is presented together with basic concepts and potential, new applications to Oncology and Integrative Cancer Biology. Novel translational oncogenomics research is rapidly expanding through the application of advanced technology, research findings and computational tools/models to both pharmaceutical and clinical problems. A self-contained presentation is adopted that covers both fundamental concepts and the most recent biomedical, as well as clinical, applications. Sample analyses in recent clinical studies have shown that gene expression data can be employed to distinguish between tumor types as well as to predict outcomes. Potentially important applications of such results are individualized human cancer therapies or, in general, ‘personalized medicine’. Several cancer detection techniques are currently under development both in the direction of improved detection sensitivity and increased time resolution of cellular events, with the limits of single molecule detection and picosecond time resolution already reached. The urgency for the complete mapping of a human cancer interactome with the help of such novel, high-efficiency / low-cost and ultra-sensitive techniques is also pointed out

Improving a fixed parameter tractability time bound for the shadow problem

AbstractConsider a forest of k trees and n nodes together with a (partial) function σ mapping leaves of the trees to non-root nodes of other trees. Define the shadow of a leaf ℓ to be the subtree rooted at σ(ℓ). The shadow problem asks whether there is a set S of leaves exactly one from each tree such that none of these leaves lies in the shadow of another leaf in S. This graph theoretical problem as shown in Franco et al. (Discrete Appl. Math. 96 (1999) 89) is equivalent to the falsifiability problem for pure implicational Boolean formulas over n variables with k occurences of the constant false as introduced in: Heusch J. Wiedermann, P. Hajek (Eds.), Proceedings of the Twentieth International Symposium on Mathematical Foundations of Computer Science (MFCS’95), Prague, Czech Republic, Lecture Notes in Computer Science, Vol. 969, Springer, Berlin, 1995, pp. 221–226, where its NP-completeness is shown for arbitrary values of k and a time bound of O(nk) for fixed k was obtained. In Franco et al. (1999) this bound is improved to O(n2kk) showing the problem's fixed parameter tractability (Congr. Numer. 87 (1992) 161). In this paper the bound O(n33k) is achieved by dynamic programming techniques thus significantly improving the fixed parameter part