Cyclic projectors and separation theorems in idempotent convex geometry

Semimodules over idempotent semirings like the max-plus or tropical semiring have much in common with convex cones. This analogy is particularly apparent in the case of subsemimodules of the n-fold cartesian product of the max-plus semiring it is known that one can separate a vector from a closed subsemimodule that does not contain it. We establish here a more general separation theorem, which applies to any finite collection of closed semimodules with a trivial intersection. In order to prove this theorem, we investigate the spectral properties of certain nonlinear operators called here idempotent cyclic projectors. These are idempotent analogues of the cyclic nearest-point projections known in convex analysis. The spectrum of idempotent cyclic projectors is characterized in terms of a suitable extension of Hilbert's projective metric. We deduce as a corollary of our main results the idempotent analogue of Helly's theorem.Comment: 20 pages, 1 figur

Detecting Features from Confusion Matrices using Generalized Formal Concept Analysis

We claim that the confusion matrices of multiclass problems can be analyzed by means of a generalization of Formal Concept Analysis to obtain symbolic information about the feature sets of the underlying classification task.We prove our claims by analyzing the confusion matrices of human speech perception experiments and comparing our results to those elicited by experts.This work has been supported by Spanish Government-Comisión Interministerial de Ciencia y Tecnología TEC2008-02473/TEC y TEC2008-06382/TEC.Publicad

Synthesizing Systems with Optimal Average-Case Behavior for Ratio Objectives

We show how to automatically construct a system that satisfies a given logical specification and has an optimal average behavior with respect to a specification with ratio costs. When synthesizing a system from a logical specification, it is often the case that several different systems satisfy the specification. In this case, it is usually not easy for the user to state formally which system she prefers. Prior work proposed to rank the correct systems by adding a quantitative aspect to the specification. A desired preference relation can be expressed with (i) a quantitative language, which is a function assigning a value to every possible behavior of a system, and (ii) an environment model defining the desired optimization criteria of the system, e.g., worst-case or average-case optimal. In this paper, we show how to synthesize a system that is optimal for (i) a quantitative language given by an automaton with a ratio cost function, and (ii) an environment model given by a labeled Markov decision process. The objective of the system is to minimize the expected (ratio) costs. The solution is based on a reduction to Markov Decision Processes with ratio cost functions which do not require that the costs in the denominator are strictly positive. We find an optimal strategy for these using a fractional linear program.Comment: In Proceedings iWIGP 2011, arXiv:1102.374

The irregular nesting problem - a new approach for nofit polygon calculation

This paper presents a new approach for generating the nofit polygon (NFP) that is simple, intuitive and computationally efficient. The NFP has in recent years become an important tool for handling the geometric calculations for two-dimensional irregular shape nesting problems. Its value lies in reducing the computational complexity of detecting whether two polygons overlap. The proposed NFP generator is based on the novel concept of trace line segments that are derived from the interaction of the two-component polygons. The complete set of the trace line segments contain all the boundary edges of the NFP and some internal points that need to be discarded. Algorithms for deriving the trace line segments, efficiently determining those segments that form the boundary of the NFP and the identification of holes and degenerate cases are describe

A Note on Tropical Triangles in the Plane

We define transversal tropical triangles (affine and projective) and characterize them via six inequalities to be satisfied by the coordinates of the vertices. We prove that the vertices of a transversal tropical triangle are tropically independent and they tropically span a classical hexagon whose sides have slopes ∞, 0, 1. Using this classical hexagon, we determine a parameter space for transversal tropical triangles. The coordinates of the vertices of a transversal tropical triangle determine a tropically regular matrix. Triangulations of the tropical plane are obtained