Finite Automata Algorithms in Map-Reduce

In this thesis the intersection of several large nondeterministic finite automata (NFA's) as well as minimization of a large deterministic finite automaton (DFA) in map-reduce are studied. We have derived a lower bound on replication rate for computing NFA intersections and provided three concrete algorithms for the problem. Our investigation of the replication rate for each of all three algorithms shows where each algorithm could be applied through detailed experiments on large datasets of finite automata. Denoting n the number of states in DFA A, we propose an algorithm to minimize A in n map-reduce rounds in the worst-case. Our experiments, however, indicate that the number of rounds, in practice, is much smaller than n for all DFA's we examined. In other words, this algorithm converges in d iterations by computing the equivalence classes of each state, where d is the diameter of the input DFA

Pre-Reduction Graph Products: Hardnesses of Properly Learning DFAs and Approximating EDP on DAGs

The study of graph products is a major research topic and typically concerns the term $f(G*H)$, e.g., to show that $f(G*H)=f(G)f(H)$. In this paper, we study graph products in a non-standard form $f(R[G*H]$ where $R$ is a "reduction", a transformation of any graph into an instance of an intended optimization problem. We resolve some open problems as applications. (1) A tight $n^{1-\epsilon}$-approximation hardness for the minimum consistent deterministic finite automaton (DFA) problem, where $n$ is the sample size. Due to Board and Pitt [Theoretical Computer Science 1992], this implies the hardness of properly learning DFAs assuming $NP\neq RP$ (the weakest possible assumption). (2) A tight $n^{1/2-\epsilon}$ hardness for the edge-disjoint paths (EDP) problem on directed acyclic graphs (DAGs), where $n$ denotes the number of vertices. (3) A tight hardness of packing vertex-disjoint $k$-cycles for large $k$. (4) An alternative (and perhaps simpler) proof for the hardness of properly learning DNF, CNF and intersection of halfspaces [Alekhnovich et al., FOCS 2004 and J. Comput.Syst.Sci. 2008]

The Covering Problem

An important endeavor in computer science is to understand the expressive power of logical formalisms over discrete structures, such as words. Naturally, "understanding" is not a mathematical notion. This investigation requires therefore a concrete objective to capture this understanding. In the literature, the standard choice for this objective is the membership problem, whose aim is to find a procedure deciding whether an input regular language can be defined in the logic under investigation. This approach was cemented as the right one by the seminal work of Sch\"utzenberger, McNaughton and Papert on first-order logic and has been in use since then. However, membership questions are hard: for several important fragments, researchers have failed in this endeavor despite decades of investigation. In view of recent results on one of the most famous open questions, namely the quantifier alternation hierarchy of first-order logic, an explanation may be that membership is too restrictive as a setting. These new results were indeed obtained by considering more general problems than membership, taking advantage of the increased flexibility of the enriched mathematical setting. This opens a promising research avenue and efforts have been devoted at identifying and solving such problems for natural fragments. Until now however, these problems have been ad hoc, most fragments relying on a specific one. A unique new problem replacing membership as the right one is still missing. The main contribution of this paper is a suitable candidate to play this role: the Covering Problem. We motivate this problem with 3 arguments. First, it admits an elementary set theoretic formulation, similar to membership. Second, we are able to reexplain or generalize all known results with this problem. Third, we develop a mathematical framework and a methodology tailored to the investigation of this problem

Completeness for Identity-free Kleene Lattices

We provide a finite set of axioms for identity-free Kleene lattices, which we prove sound and complete for the equational theory of their relational models. Our proof builds on the completeness theorem for Kleene algebra, and on a novel automata construction that makes it possible to extract axiomatic proofs using a Kleene-like algorithm

Dense urban elevation models from stereo images by an affine region merging approach

The main subject of this thesis is the computation of Dense Disparity Maps from a pair of satelite or aerial stereo images from an urban scene, taken from two different viewpoints. Several steps are needed to obtain the ?nal disparity map from the pair of images. We focus here on one of these steps: how to match the points in one image with the points in the other one. This matching process is closely related to the computation of the altitudes of the objects present in the scene. Indeed, the precision we can obtain in these altitude values is directly proportional to the precision in the matching process. This precision in the altitude is also inversely proportional to the distance between both viewpoints where the images are taken(baseline). The matching process is a widely studied ?eld in the Computer Vision Community and several methods and algorithms have been developed so far ([31, 27, 49]). Most of them consider a big base- line con?guration, which increases the performance in the altitude and also simpli?es the matching process. However, this assumption presents a major drawback with objects that are occluded in one image but appear in the other one. The bigger the baseline is, the more objects are occluded in one image and are not occluded in the other one. Recently, a different approach in which the images are taken with a very small baseline started to be analyzed ([19, 20]). This approach has the advantage of eliminating most of the ambiguities presented when one object occluded in one image is not occluded in the other one. Indeed, if we consider that we have a very small baseline, the occlusions presented in both images are almost the same. Now, this con?guration obviously decreases the precision in the ?nal altitude. In order to continue obtaining highly accurate altitude values, the precision in the matching process must be im- proved. The methods developed so far which consider the small baseline approach, compute altitude values with a high precision at some points, but leave the rest of them with no altitude values at all, generating a non-dense disparity map. Based on the fact that piecewise-a?ne models are reasonable for the elevation in urban areas, we propose a new method to interpolate and denoise those non-dense disparity maps. Under lambertian illumination hypothesis 1 , it is reasonable to assume that homogeneous regions in the graylevel image, correspond to the same a?ne elevation model. In other words, the borders between the piecewise a?ne elevation model are included to a large extent within contrasted graylevel borders. Hence, it is reasonable to look for an piecewise a?ne ?t to the elevation model where the borders between regions are taken from a graylevel segmenation of the image We present a region-merging algorithm that starts with an over-segmentation of the gray-level im- age. The disparity values at each region are approximated by an a?ne model, and a meaningfulness measure of the ?t is assigned to each of them. Using this meaningfulness as a merging order, the method iterates until no new merge is possible, according to a merging criterion which is also based on the meaningfulness of each pair of neighboring regions. In the last step, the algorithm performs a validation of the ?nal regions using again the meaningfulness of the ?t. The regions validated in this last step are those for which the a?ne model is a good approximation. The region-merging algorithm presented in this work can be seen as an attempt to incorporate a semantical meaning to real scenes: we have developed a validation method to determine whether the data within a region is well approximated by an a?ne model or not. Hence, we could analyze more complex models, de?ning a suitable validation criterion for each of them. In this way, we can search for the model that best explains a given data set in terms of its meaningfulness

Regular Separability of Parikh Automata

We investigate a subclass of languages recognized by vector addition systems, namely languages of nondeterministic Parikh automata. While the regularity problem (is the language of a given automaton regular?) is undecidable for this model, we surprisingly show decidability of the regular separability problem: given two Parikh automata, is there a regular language that contains one of them and is disjoint from the other? We supplement this result by proving undecidability of the same problem already for languages of visibly one counter automata

Quantum automata, braid group and link polynomials

The spin--network quantum simulator model, which essentially encodes the (quantum deformed) SU(2) Racah--Wigner tensor algebra, is particularly suitable to address problems arising in low dimensional topology and group theory. In this combinatorial framework we implement families of finite--states and discrete--time quantum automata capable of accepting the language generated by the braid group, and whose transition amplitudes are colored Jones polynomials. The automaton calculation of the polynomial of (the plat closure of) a link L on 2N strands at any fixed root of unity is shown to be bounded from above by a linear function of the number of crossings of the link, on the one hand, and polynomially bounded in terms of the braid index 2N, on the other. The growth rate of the time complexity function in terms of the integer k appearing in the root of unity q can be estimated to be (polynomially) bounded by resorting to the field theoretical background given by the Chern-Simons theory.Comment: Latex, 36 pages, 11 figure

Detection of traffic congestion and incidents from GPS trace analysis

This paper presents an expert system for detecting traffic congestion and incidents from real-time GPS data collected from GPS trackers or drivers’ smartphones. First, GPS traces are pre-processed and placed in the road map. Then, the system assigns to each road segment of the map a traffic state based on the speeds of the vehicles. Finally, it sends to the users traffic alerts based on a spatiotemporal analysis of the classified segments. Each traffic alert contains the affected area, a traffic state (e.g., incident, slowed traffic, blocked traffic), and the estimated velocity of vehicles in the area. The proposed system is intended to be a valuable support tool in traffic management for municipalities and citizens. The information produced by the system can be successfully employed to adopt actions for improving the city mobility, e.g., regulate vehicular traffic, or can be exploited by the users, who may spontaneously decide to modify their path in order to avoid the traffic jam. The elaboration performed by the expert system is independent of the context (urban o non-urban) and may be directly employed in several city road networks with almost no change of the system parameters, and without the need for a learning process or historical data. The experimental analysis was performed using a combination of simulated GPS data and real GPS data from the city of Pisa. The results on incidents show a detection rate of 91.6%, and an average detection time lower than 7 min. Regarding congestion, we show how the system is able to recognize different levels of congestion depending on different road use