Accepting grammars and systems

We investigate several kinds of regulated rewriting (programmed, matrix, with regular control, ordered, and variants thereof) and of parallel rewriting mechanisms (Lindenmayer systems, uniformly limited Lindenmayer systems, limited Lindenmayer systems and scattered context grammars) as accepting devices, in contrast with the usual generating mode. In some cases, accepting mode turns out to be just as powerful as generating mode, e.g. within the grammars of the Chomsky hierarchy, within random context, regular control, L systems, uniformly limited L systems, scattered context. Most of these equivalences can be proved using a metatheorem on so-called context condition grammars. In case of matrix grammars and programmed grammars without appearance checking, a straightforward construction leads to the desired equivalence result. Interestingly, accepting devices are (strictly) more powerful than their generating counterparts in case of ordered grammars, programmed and matrix grammars with appearance checking (even programmed grammarsm with unconditional transfer), and 1lET0L systems. More precisely, if we admit erasing productions, we arrive at new characterizations of the recursivley enumerable languages, and if we do not admit them, we get new characterizations of the context-sensitive languages. Moreover, we supplement the published literature showing: - The emptiness and membership problems are recursivley solvable for generating ordered grammars, even if we admit erasing productions. - Uniformly limited propagating systems can be simulated by programmed grammars without erasing and without appearance checking, hence the emptiness and membership problems are recursively solvable for such systems. - We briefly discuss the degree of nondeterminism and the degree of synchronization for devices with limited parallelism

Mehrfach-limitierte Lindenmayer-Systeme

The theory of L systems originated with the biologist and mathematician Aristide Lindenmayer. His original goal was to provide mathematical models for the simultaneous development of cells in filamentous organisms. Since L systems may be viewed as rewriting systems, their generated languages, i.e., sets of organisms encoded by strings, are also subject to formal language theory, which aims to classify formal languages as well as their generating mechanisms according to various properties, such as generative power, decidability, etc. D. Wätjen introduced and studied k-limited L systems in order to combine the purely sequential mode of rewriting and the purely parallel mode of rewriting in context-free grammars, respectively, L systems. In biology, these systems may be interpreted as organisms, for which the simultaneous growth of cells is restricted by the supply of some resources of food being limited by some finite value k. In this thesis the constraint of a common limit k is relaxed in favor of individual resource limits k(a) for every cell-type a, which yields the new notion of multi-limited L system. The language families generated by such systems are then classified according to their sets of limits k(a). At first, an intuitive approach to the different mechanisms of the L system variants is provided by presenting a method for the graphical interpretation of L systems, the so-called turtle interpretation. Suitable computer programs implementing a turtle interpreter as well as free-programmable simulators for multi-limited, k-limited, and uniformly k-limited L systems, are developed and their source-code is appended. Subsequently, language families generated by multi-limited L systems are compared to each other, to Wätjen's k-limited as well as to non-limited language families, and to the families of the Chomsky Hierarchy. Besides asymptotically comparing the generative power of multi-limited L systems to that of the underlying non-limited L systems, also their closure properties are investigated.Der Biologe und Mathematiker Aristide Lindenmayer begründete die Theorie der L-Systeme. Das ursprüngliche Ziel dieser Theorie ist die Bereitstellung mathematischer Modelle zur Untersuchung des simultanen Zellwachstums fadenartiger Organismen. Da L-Systeme als eine Art von Ersetzungssystemen definiert sind, sind ihre erzeugten Sprachen, d.h. die Mengen der durch Zeichenketten beschriebenen Organismen, ebenfalls Gegenstand der Theorie der formalen Sprachen. Diese Theorie klassifiziert formale Sprachen sowie ihre Erzeugungsmechanismen gemäß ihrer Eigenschaften, wie z.B. Erzeugungsmächtigkeit oder Entscheidbarkeit. Als ein Sprachen-erzeugender Mechanismus, der zwischen der rein sequentiellen Ersetzung kontextfreier Grammatiken und der rein parallelen Ersetzung von L-Systemen liegt, sind k-limitierte L-Systeme von D. Wätjen eingeführt und untersucht worden. In der Biologie können diese Systeme als Organismen interpretiert werden, deren simultanes Zellwachstum beschränkt ist durch individuelle Nahrungsvorräte mit einer einheitlichen endlichen Kapazität k. Die in dieser Arbeit betrachteten mehrfach-limitierten L-Systeme bilden eine Verallgemeinerung der k-limitierten L-Systeme, indem sie für jeden Zelltyp a einen individuellen Nahrungsvorrat mit einer spezifischen Kapazität k(a) anstelle der einheitlichen Kapazität k vorsehen. Diese Arbeit führt mehrfach-limitierte L-Systeme ein und definiert eine geeignete Kategorisierung der von ihnen erzeugten Sprachfamilien anhand der erlaubten Mengen von Limits k(a). Zunächst wird ein intuitiver Zugang zu den verschiedenen Mechanismen der L-System-Varianten ermöglicht, indem eine Methode zur grafischen Interpretation von L-Systemen, die sogenannte Turtle-Interpretation, vorgestellt wird. Hierzu werden geeignete Computer-Programme für einen Turtle-Interpreter sowie für frei programmierbare Simulatoren von mehrfach-limitierten, k-limitierten sowie uniform k-limitierten L-Systemen erstellt und ihr Quell-Code zur Verfügung gestellt. Die von mehrfach-limitierten L-Systemen erzeugten Sprachfamilien werden bzgl. ihrer Inklusionseigenschaften untereinander, mit Wätjens k-limitierten Sprachfamilien, mit den nicht-limitierten Sprachfamilien sowie mit der Chomsky Hierarchie verglichen. Die Erzeugungsmächtigkeit von mehrfach-limitierten L-Systemen wird asymptotisch verglichen mit den jeweils unterliegenden nicht-limitierten L-Systemen. Des weiteren werden die Abschlusseigenschaften der mehrfach-limitierten L-Systeme untersucht

Membership for limited ET0L languages is not decidable

In this paper, we show how to encode arbitrary enumerable set of numbers given by register machines within limited EPT0L systems and programmed grammars with unconditional transfer.This result has various consequences, e.g.the existence of nonrecursive sets generable by 1lET0L systems or by programmed grammars with unconditional transfer. Moreover, ordered grammars are strictly less powerful than 1lET0L systems

Contributions of formal language theory to the study of dialogues

For more than 30 years, the problem of providing a formal framework for modeling dialogues has been a topic of great interest for the scientific areas of Linguistics, Philosophy, Cognitive Science, Formal Languages, Software Engineering and Artificial Intelligence. In the beginning the goal was to develop a "conversational computer", an automated system that could engage in a conversation in the same way as humans do. After studies showed the difficulties of achieving this goal Formal Language Theory and Artificial Intelligence have contributed to Dialogue Theory with the study and simulation of machine to machine and human to machine dialogues inspired by Linguistic studies of human interactions. The aim of our thesis is to propose a formal approach for the study of dialogues. Our work is an interdisciplinary one that connects theories and results in Dialogue Theory mainly from Formal Language Theory, but also from another areas like Artificial Intelligence, Linguistics and Multiprogramming. We contribute to Dialogue Theory by introducing a hierarchy of formal frameworks for the definition of protocols for dialogue interaction. Each framework defines a transition system in which dialogue protocols might be uniformly expressed and compared. The frameworks we propose are based on finite state transition systems and Grammar systems from Formal Language Theory and a multi-agent language for the specification of dialogue protocols from Artificial Intelligence. Grammar System Theory is a subfield of Formal Language Theory that studies how several (a finite number) of language defining devices (language processors or grammars) jointly develop a common symbolic environment (a string or a finite set of strings) by the application of language operations (for instance rewriting rules). For the frameworks we propose we study some of their formal properties, we compare their expressiveness, we investigate their practical application in Dialogue Theory and we analyze their connection with theories of human-like conversation from Linguistics. In addition we contribute to Grammar System Theory by proposing a new approach for the verification and derivation of Grammar systems. We analyze possible advantages of interpreting grammars as multiprograms that are susceptible of verification and derivation using the Owicki-Gries logic, a Hoare-based logic from the Multiprogramming field

Structured Tensor Recovery and Decomposition

Tensors, a.k.a. multi-dimensional arrays, arise naturally when modeling higher-order objects and relations. Among ubiquitous applications including image processing, collaborative filtering, demand forecasting and higher-order statistics, there are two recurring themes in general: tensor recovery and tensor decomposition. The first one aims to recover the underlying tensor from incomplete information; the second one is to study a variety of tensor decompositions to represent the array more concisely and moreover to capture the salient characteristics of the underlying data. Both topics are respectively addressed in this thesis. Chapter 2 and Chapter 3 focus on low-rank tensor recovery (LRTR) from both theoretical and algorithmic perspectives. In Chapter 2, we first provide a negative result to the sum of nuclear norms (SNN) model---an existing convex model widely used for LRTR; then we propose a novel convex model and prove this new model is better than the SNN model in terms of the number of measurements required to recover the underlying low-rank tensor. In Chapter 3, we first build up the connection between robust low-rank tensor recovery and the compressive principle component pursuit (CPCP), a convex model for robust low-rank matrix recovery. Then we focus on developing convergent and scalable optimization methods to solve the CPCP problem. In specific, our convergent method, proposed by combining classical ideas from Frank-Wolfe and proximal methods, achieves scalability with linear per-iteration cost. Chapter 4 generalizes the successive rank-one approximation (SROA) scheme for matrix eigen-decomposition to a special class of tensors called symmetric and orthogonally decomposable (SOD) tensor. We prove that the SROA scheme can robustly recover the symmetric canonical decomposition of the underlying SOD tensor even in the presence of noise. Perturbation bounds, which can be regarded as a higher-order generalization of the Davis-Kahan theorem, are provided in terms of the noise magnitude

Scheduling multicasts on unit-capacity trees and meshes

This paper studies the multicast routing and admission control problem on unit-capacity tree and mesh topologies in the throughput-model. The problem is a generalization of the edge-disjoint paths problem and is NP-hard both on trees and meshes. We study both the offline and the online version of the problem: In the offline setting, we give the first constant-factor approximation algorithm for trees, and an O((log log n)^2)-factor approximation algorithm for meshes. In the online setting, we give the first polylogarithmic competitive online algorithm for tree and mesh topologies. No polylogarithmic-competitive algorithm is possible on general network topologies [Bartal,Fiat,Leonardi, 96], and there exists a polylogarithmic lower bound on the competitive ratio of any online algorithm on tree topologies [Awerbuch,Azar,Fiat,Leighton, 96]. We prove the same lower bound for meshes

Ogden's lemma for random permitting and forbidding context picture languages and table-driven context-free picture languages

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. Johannesburg, February 16, 2015.Random context picture grammars are used to generate pictures through successive refinement. There are three important subclasses of random context picture grammars, namely random permitting context picture grammars, random forbidding context picture grammars and table-driven context-free picture grammars. These grammars generate the random permitting context picture languages, random forbidding context picture languages and table-driven context-free picture languages, respectively. Theorems exist which provide necessary conditions that have to be satisfied by a language before it can be classified under a particular subclass. Some of these theorems include the pumping and shrinking lemmas, which have been developed for random permitting context picture languages and random forbidding context picture languages respectively. Two characterization theorems were developed for the table-driven context-free picture languages. This dissertation examines these existing theorems for picture languages, i.e., the pumping and shrinking lemmas and the two characterisation theorems, and attempts to prove theorems, which will provide an alternative to the existing theorems and thus provide new tools for identifying languages that do not belong to the various classes. This will be done by adapting Ogden’s idea of marking parts of a word which was done for the string case. Our theorems essentially involve marking parts of a picture such that the pumping operation increases the number of marked symbols and the shrinking operation reduces it

Learning Tuple Probabilities in Probabilistic Databases

Learning the parameters of complex probabilistic-relational models from labeled training data is a standard technique in machine learning, which has been intensively studied in the subfield of Statistical Relational Learning (SRL), but---so far---this is still an under-investigated topic in the context of Probabilistic Databases (PDBs). In this paper, we focus on learning the probability values of base tuples in a PDB from query answers, the latter of which are represented as labeled lineage formulas. Specifically, we consider labels in the form of pairs, each consisting of a Boolean lineage formula and a marginal probability that comes attached to the corresponding query answer. The resulting learning problem can be viewed as the inverse problem to confidence computations in PDBs: given a set of labeled query answers, learn the probability values of the base tuples, such that the marginal probabilities of the query answers again yield in the assigned probability labels. We analyze the learning problem from a theoretical perspective, devise two optimization-based objectives, and provide an efficient algorithm (based on Stochastic Gradient Descent) for solving these objectives. Finally, we conclude this work by an experimental evaluation on three real-world and one synthetic dataset, while competing with various techniques from SRL, reasoning in information extraction, and optimization