J. Andres Montoya

Linear network coding and the model theory of linear rank inequalities

Let n ≥ 4. Can the entropic region of order n be defined by a finite list of polynomial inequalities? This question was first asked by Chan and Grant. We showed, in a companion paper, that if it were the case one could solve many algorithmic problems coming from network coding, index coding and secret sharing. Unfortunately, it seems that the entropic regions are not semialgebraic. Are the Ingleton regions semialgebraic sets? We provide some evidence showing that the Ingleton regions are semialgebraic. Furthermore, we show that if the Ingleton regions are semialgebraic, then one can solve many algorithmic problems coming from Linear Network Coding.

On the complexity of sandpile critical avalanches

In this work, we study The Abelian Sandpile Model from the point of view of computational complexity. We begin by studying the length distribution of sandpile avalanches triggered by the addition of two critical configurations: we prove that those avalanches are long on average, their length is bounded below by a constant fraction of the length of the longest critical avalanche which is, in most of the cases, superlinear. At the end of the paper we take the point of view of computational complexity, we analyze the algorithmic hardness of the problem consisting in computing the addition of two critical configurations, we prove that this problem is P complete, and we prove that most algorithmic problems related to The Abelian Sandpile Model are NC reducible to it.

On the Complexity of Sandpile Prediction Problems

In this work we study the complexity of Sandpile prediction problems on several classes of directed graphs. We focus our research on low-dimensional directed lattices. We prove some upper and lower bounds for those problems. Our approach is based on the analysis of some reachability problems related to sandpiles.

On a Conjecture by Christian Choffrut

It is one of the most famous open problems to determine the minimum amount of states required by a deterministic finite automaton to distinguish a pair of strings, which was stated by Christian Choffrut more than thirty years ago. We investigate the same question for different automata models and we obtain new upper and lower bounds for some of them including alternating, ultrametric, quantum, and affine finite automata.

On the Real-state Processing of Regular Operations and The Sakoda-Sipser Problem

In this work we study some aspects of state-complexity related to the very famous Sakoda-Sipser problem. We study the state-complexity of the regular operations, we survey the known facts and, by the way, we find some new and simpler proofs of some well known results. The analysis of the state of art allowed us to find a new and meaningful notion: Real-state processing. We investigate this notion, looking for a model of deterministic finite automata holding such an interesting property. We establish some preliminary results, which seem to indicate that there does not exists a model of deterministic finite automata having realstate processing of regular expressions, but, on the other hand, we are able of exhibiting a deterministic model of finite automata having real-state processing of star free regular expressions.

On discerning strings with finite automata

We study the problem of discerning strings with deterministic finite state automata (DFAs, for short). We begin with a survey on the historical and algorithmic roots of this problem. Then, we focus on the maximun number of states that are necessary to separate two strings of a given length. We survey the most important results concerning this issue and we study the problem from the point of view of some alternative models of automata. The preliminary results concerning the last issue motivate us to formulate a conjecture stating that DFAs can separate any pair of strings by using a logarithmic number of states. We give some evidence supporting our conjecture.

Network coding and the model theory of linear information inequalities

Let n ≥ 4, can the entropic region of order n be defined by a finite list of polynomial inequalities? This question was first asked by Chan and Grant. We show that if it were the case one could solve many algorithmic problems coming from Network Coding, Index Coding and Secret Sharing. Unfortunately, it seems that the entropic regions of order larger than four are not semialgebraic. Actually, we guess that it is the case and we provide strong evidence supporting our conjecture.

On the Synchronization of Planar Automata.

Planar automata seems to be representative of the synchronizing behavior of deterministic finite state automata. We conjecture that Cerny’s conjecture holds true, if and only if, it holds true for planar automata. We provide new (and old) evidence concerning the conjectured C erny-universality of planar automata.

On the linear algebra of biological homochirality

We look for structural properties of chemical networks giving place to homochiral phenomena. We found a necessary condition for homochirality that we call Frank inequality, and which is a linear inequality related to the entries of the jacobian matrices that occur at racemic steady states. We also investigate the existence of stronger conditions that can be formulated in a similar algebraic way. Those investigations lead us to introduce a homochirality degree for the racemic states of chiral neworks, which is intended to measure the probability of observing homochiral dynamics after perturbing those states. It is important to stress that all the introduced concepts and degrees are effective. The later fact allows us to develop an algorithm that can be used to, given a chiral network as input, compute large samples of steady states of different degrees.

On the information rates of homomorphic secret sharing schemes

Abstract Homomorphic secret sharing schemes have been of fundamental importance in the development of secure multyparty protocols, like for example e-voting protocols. We propose a classification of homorphic schemes between three categories: Linear, abelian and nonabelian schemes. We study the information rates that can be achieved by those three types of schemes. We prove that nonabelian schemes outperform linear schemes, and we conjecture that the same is true for abelian schemes. We provide some strong evidence concerning the conjecture.