Intercalation properties of context-free languages

Context-freedom of a language implies certain intercalation properties known as pumping or iteration lemmas. Although the question of a converse result for some of the properties has been studied, it is still not entirely clear how these properties are related, which are the stronger ones and which are weaker;Among the intercalation properties for context-free languages the better known are the general pumping conditions (generalized Ogden\u27s, Ogden\u27s and classic pumping conditions), Sokolowski-type conditions (Sokolowski\u27s and Extended Sokolowski\u27s conditions) and the Interchange condition. We present a rather systematic investigation of the relationships among these properties; it turns out that the three types of properties, namely pumping, Sokolowski-type and interchange, above are independent. However, the interchange condition is strictly stronger than the Sokolowski\u27s condition;Intercalation properties of some subclasses of context-free languages are also studied. We prove a pumping lemma and an Ogden\u27s lemma for nonterminal bounded languages and show that none of these two conditions is sufficient. We also investigate three of Igarashi\u27s pumping conditions for real-time deterministic context-free languages and show that these conditions are not sufficient either. Furthermore, we formulate linear analogues of the general pumping and interchange conditions and then compare them to the general context-free case. The results show that these conditions are also independent

Logic and information: a unifying approach to semantic information theory

Die renommierte Informationstheorie der Informatik, die auf Shannon zurückgeht, beschäftigt sich fast ausschliesslich mit dem Messen des Informationsgehalts von Nachrichten. Das blosse Messen sagt jedoch wenig über die Beschaffenheit von Information aus. Die Frage, was Information wirklich ist, bleibt offen. Deshalb verfolgt diese Dissertation zwei Hauptziele: Zum einen soll eine adäquate Definition von Information auf konzeptueller Ebene gegeben werden. Diese Definition beruht auf einem abstrakten, axiomatischen Framework, genannt Informationsalgebra, das aus einem semantischen Blickwinkel betrachtet wird. Daraus resultiert eine algebraische Theorie semantischer Information, die von Logik veranschaulicht wird. Zum anderen soll die algebraische Theorie semantischer Information validiert werden. Dazu wurden drei semantische Informationstheorien aus anderen Disziplinen ausgewählt, dargestellt und mit der algebraischen Theorie semantischer Information verglichen. Logik dient als Beispiel für alle vier Theorien. Die algebraische Theorie semantischer Information Eine Informationsalgebra ist eine zweisortige Struktur, die aus einer Menge von möglichen Informationen und einem Verband von Fragen besteht sowie aus zwei Operationen (Kombination und Fokussierung), die fünf Axiome erfüllen. Information hat Eigenschaften, die mit Hilfe des Informationsalgebra-Frameworks formalisiert werden können: Information bezieht sich auf Fragen, Fragen wiederum stehen untereinander in einer gewissen Beziehung. Um Information zu extrahieren, wird sie auf eine bestimmte Frage fokussiert. Da Information meist aus verschiedenen Quellen stammt und teilweise unvollständig ist, muss sie kombiniert werden um einen Gesamteindruck zu vermitteln. Information kann auf zwei verschiedene Arten betrachtet werden. Das Interesse kann entweder auf die Darstellung oder die Bedeutung der Information gerichtet sein. Zur Darstellung der Information wird eine (formale) Sprache eingesetzt. Die Auseinandersetzung mit der Bedeutung der Information bedarf jedoch der Semantik, die Gegenstand dieser Dissertation ist. Die semantische Auslegung des Informationsalgebra- Framework erlaubt folgende Rückschlüsse über die Beschaffenheit von Information und Fragen: Semantische Information ist durch eine Menge von verschiedenen Möglichkeiten gegeben. Information wird als Antwort auf eine (implizite) Frage wahrgenommen. Eine Frage wird semantisch durch mögliche Antworten beschrieben. Diese Erkenntnisse führen zur algebraischen Theorie semantischer Information, die auf viele Formalismen, u. a. auf Logik, angewandt werden kann. Es wird gezeigt, dass Aussagenlogik und Prädikatenlogik Informationsalgebren bilden. Da das Gewicht auf der Bedeutung von Information liegt, werden die dazugehörigen Beweise auf semantischer Ebene ausgeführt. Auch in anderen Disziplinen, die Logik einsetzen, wie z. B. in der Philosophie oder der Sprachwissenschaft, finden sich semantische Informationstheorien. Drei dieser Theorien werden in dieser Dissertation vorgestellt. 1. In den frühen 50er Jahren des 20. Jahrhunderts entwickelten Carnap und Bar-Hillel eine semantische Informationstheorie, die sich sehr grundlegend mit semantischer Information und ihrer Messung beschäftigt. Als Beispiel dient ihnen eine beschränkte, monadische prädikatenlogische Sprache. 2. In den frühen 80er Jahren des 20. Jahrhunderts haben Groenendijk und Stokhof eine Theorie über die Semantik von Fragen und die Pragmatik von Antworten entwickelt. Diese Theorie wurde zu Beginn des 21. Jahrhunderts von van Rooij erweitert. Sie setzt eine Frage mit ihren möglichen Antworten gleich und bietet somit auch eine detaillierte Beschreibung der Beschaffenheit von Antworten. Aussagenlogik und Prädikatenlogik werden im Rahmen von Beispielen betrachtet. 3. In den späten 90er Jahren des 20. Jahrhunderts entstand die Theorie des Informationsflusses von Barwise und Seligman, die hauptsächlich auf Prädikatenlogik angewandt wird. Die Theorie des Informationsflusses dient der Darstellung von Information und ihres Transports innerhalb von verteilten Systemen. Barwise und Seligman haben eine duale Sichtweise von Information, die sowohl Syntax als auch Semantik berücksichtigt. Ihre Vorgehensweise wird in dieser Dissertation mit dem Ansatz der formalen Konzeptanalyse aus den 80er Jahren in Verbindung gebracht. Das Informationsalgebra-Framework ist eine Abstraktion dieser drei Theorien. Aus diesem Grund stellt die algebraische Theorie semantischer Information einen allumfassenden Ansatz für die drei Theorien dar.The commonly used information theory, going back to Shannon, is almost exclusively concerned with the measure of information. However, by measuring information, one does not get very much to know about its nature, about what information actually is. Therefore, this thesis has two main goals: The first is to provide an adequate definition of the concept of information, by a semantic interpretation of the abstract, axiomatic information algebra framework. This leads to the formulation of an algebraic theory of semantic information, which is exemplified by logics. The second is to validate this theory, by comparison with already established semantic information theories (which also apply to logics) of other disciplines. An information algebra is a two-sorted algebra, consisting of a set of possible pieces of information, of a lattice of questions and of two operations (combination and focusing), that satisfy a set of five axioms. The following properties of information and questions are formalized by the information algebra framework: pieces of information refer to related questions; pieces of information can be focused, in order to extract information relative to some specific question of interest; pieces of information may be combined (aggregated). Information can be perceived in two ways. One can look at how information is represented, or one can examine what information expresses. Information representation involves a (formal) language. In order to understand the meaning of the information, semantics is needed. In this thesis, we have chosen the latter approach. A semantic interpretation of the information algebra framework allows drawing conclusions about the nature of information and questions: A semantic piece of information is a set of possibilities, which may be interpreted in different ways. A piece of information is perceived as an answer to a question. A question, in turn, is semantically given by its possible answers. These results constitute the algebraic theory of semantic information, which applies to many instances, including logics. Propositional logic and predicate logic are shown to be information algebra instances. Since the meaning of information matters, the proofs are given on the semantic level. In disciplines which are also dealing with logics, like philosophy or linguistics, semantic information theories can be found, too. Three of them are presented in this thesis. 1. Carnap and Bar-Hillel's theory of semantic information from the early 1950s provides a very basic framework for semantic information and its measure. Information is perceived as a set of excluded possibilities. The theory is instantiated by a restricted monadic predicate logic language. 2. Groenendijk and Stokhof's theory of the semantics of questions and the pragmatics of answers provides a framework for questions. This theory has been developed in the early 1980s and has been extended by van Rooij in the first decade of the 21st century. As questions are identified with their possible answers, a detailed description of the nature of answers is also given. Two instances, propositional and predicate logic, are considered. 3. Barwise and Seligman's theory of information flow, which came up in the late 1990s, provides a framework for the representation of information and the computation with it in distributed systems. As to the representation, information is seen from a dual perspective, taking into account syntax and semantics. This dual representation is identified in this thesis with the approach of formal concept analysis, which was introduced in the 1980s. Barwise and Seligman mainly exemplify their theory by predicate logic. All these three theories fit into the information algebra framework. Therefore, the algebraic theory of semantic information encompasses these theories

Tableau Algorithms for Categorial Deduction and Parsing

Institute for Communicating and Collaborative SystemsIn this thesis we develop automated dedution mehanisms designed to keep complexity of categorial parsing under control while preserving the levels of uniformity and coverage one finds in labeled dedutive systems. First,we define the hierarhy of caluli whose computational treatment is addresed in the thesis,review the main issues and linguistic motivations behind proof-theoretical features of each calculus and describe the correspondence between proofs and semantic interpretation with respect to lambda terms. Next we introduce the rules and algorithms of a deductive system based on analytic tableaux which covers the whole hierarchy of categorial calculi presented.Completenes and termination results are shown. We then impose syntactic constraints on the calculi and elaborate label unification proceedures aimed at limiting the system's complexity. Alternative proof-search strategies are discussed and a technique for recovering syntactic structure from tableau derivations is developed. In the last chapters we compare our system with other methods used in categorial deduction,discuss design issues,heuristics and extensions,and link categorial deduction with theorem proving in recently developed logics of information flow such as channel theory

Vagueness in mathematics talk

The Cockcroft Report claimed that "mathematics provides a means of communication which is powerful, concise and unambiguous". Such precision in language may be a conventional aim of mathematics, particularly when communicated in writing. Nonetheless, as this thesis demonstrates, vagueness is commonplace when people talk about mathematics. In this thesis, I examine the circumstances in which vagueness arises in mathematics talk, and consider the practical purposes which speakers achieve by means of vague utterances in this context. The empirical database, which is considered in Chapters 4 to 7, consists almost entirely of transcripts of mathematical conversations between adult interviewers (including myself) and one or two children. The data were collected from clinical interviews focused on a small number of tasks, and from fragments of teaching. For the most part, the pupils involved in the study were aged between 9 and 12, although the age-range in Chapter 7 extends from 4 to 25. I draw on a number of approaches to discourse associated with 'pragmatics' -a field of linguistics - to analyse the motives and communicative effectiveness of speakers who deploy vagueness in mathematics talk. I claim that, for these speakers, vagueness fulfills a number of purposes, especially 'shielding', i. e. self-protection against accusation of being wrong. Another purpose is to give approximate information; sometimes to achieve shielding, but also to provide the level of detail that is deemed to be appropriate in a given situation. A different purpose, associated with a particular form of vagueness (of reference), is to compensate for lexical gaps in pursuit of effective communication of concepts and ideas. I show, in particular, how speakers use the pronouns 'it' and 'you' in mathematics talk to communicate concepts and generalisations. Some consideration is given to the intentions of 'expert speakers of mathematics when they deploy vague language. Their purposes include some of those identified for novices. Teachers also use vagueness as a means of indirectness in addressing pupils; this strategy is associated with the redress of 'face threatening acts'. My thesis is that vagueness can be viewed and presented, not as a disabling feature of language, but as a subtle and versatile device which speakers can and do deploy to make mathematical assertions with as much precision, accuracy or as much confidence as they judge is warranted by both the content and the circumstances of their utterances. I report on the validation and generalisation of my findings by an Informal Research Group of school teachers, who transcribed and analysed their own classroom interactions using the methods I had developed