Implicit learning of recursive context-free grammars

Context-free grammars are fundamental for the description of linguistic syntax. However, most artificial grammar learning experiments have explored learning of simpler finite-state grammars, while studies exploring context-free grammars have not assessed awareness and implicitness. This paper explores the implicit learning of context-free grammars employing features of hierarchical organization, recursive embedding and long-distance dependencies. The grammars also featured the distinction between left- and right-branching structures, as well as between centre- and tail-embedding, both distinctions found in natural languages. People acquired unconscious knowledge of relations between grammatical classes even for dependencies over long distances, in ways that went beyond learning simpler relations (e.g. n-grams) between individual words. The structural distinctions drawn from linguistics also proved important as performance was greater for tail-embedding than centre-embedding structures. The results suggest the plausibility of implicit learning of complex context-free structures, which model some features of natural languages. They support the relevance of artificial grammar learning for probing mechanisms of language learning and challenge existing theories and computational models of implicit learning

On the computational complexity of algebraic numbers : the Hartmanis-Stearns problem revisited

International audience— We consider the complexity of integer base expansions of algebraic irrational numbers from a computational point of view. We show that the Hartmanis–Stearns problem can be solved in a satisfactory way for the class of multistack machines. In this direction, our main result is that the base-b expansion of an algebraic irrational real number cannot be generated by a deterministic pushdown automaton. We also confirm an old claim of Cobham proving that such numbers cannot be generated by a tag machine with dilation factor larger than one

Counting, Adding, and Regular Languages

In this thesis we consider two mostly disjoint topics in formal language theory that both involve the study and use of regular languages. The first topic lies in the intersection of automata theory and additive number theory. We introduce a method of producing results in additive number theory, relying on theorem-proving software and an approximation technique. As an example of the method, we prove that every natural number greater than 25 can be written as the sum of at most 3 natural numbers whose canonical base-2 representations have an equal number of 0's and 1's. We prove analogous results about similarly defined sets using the automata theory approach, but also give proofs using more "traditional" approaches. The second topic is the study languages defined by criteria involving the number of occurrences of a particular pair of words within other words. That is, we consider languages of words z defined with respect to words x, y where z has the same number of occurrences (resp., fewer occurrences), (resp., fewer occurrences or the same number of occurrences) of x as a subword of z and y as a subword of z. We give a necessary and sufficient condition on when such languages are regular, and show how to check this condition efficiently. We conclude by briefly considering ideas tying the two topics together

Finite-State Genericity : on the Diagonalization Strength of Finite Automata

Algorithmische Generizit¨atskonzepte spielen eine wichtige Rolle in der Berechenbarkeitsund Komplexit¨atstheorie. Diese Begriffe stehen in engem Zusammenhang mit grundlegenden Diagonalisierungstechniken, und sie wurden zur Erzielung starker Trennungen von Komplexit¨atsklassen verwendet. Da f¨ur jedes Generizit¨atskonzept die zugeh¨origen generischen Mengen eine co-magere Klasse bilden, ist die Analyse generischer Mengen ein wichtiges Hifsmittel f¨ur eine quantitative Analyse struktureller Ph¨anomene. Typischerweise werden Generizit¨atskonzepte mit Hilfe von Erweiterungsfunktionen definiert, wobei die St¨arke eines Konzepts von der Komplexit¨at der zugelassenen Erwiterungsfunktionen abh¨angt. Hierbei erweisen sich die sog. schwachen Generizit¨atskonzepte, bei denen nur totale Erweiterungsfunktionen ber¨ucksichtigt werden, meist als wesentlich schw¨acher als die vergleichbaren allgemeinen Konzepte, bei denen auch partielle Funktionen zugelassen sind. Weiter sind die sog. beschr¨ankten Generizit¨atskonzepte – basierend auf Erweiterungen konstanter L¨ange – besonders interessant, da hier die Klassen der zugeh¨origen generischen Mengen nicht nur co-mager sind sondern zus¨atzlich Maß 1 haben. Generische Mengen diesen Typs sind daher typisch sowohl im topologischen wie im maßtheoretischen Sinn. In dieser Dissertation initiieren wir die Untersuchung von Generizit¨at im Bereich der Theorie der Formalen Sprachen: Wir f¨uhren finite-state-Generizit¨atskonzepte ein und verwenden diese, um die Diagonalisierungsst¨arke endlicher Automaten zu erforschen. Wir konzentrieren uns hierbei auf die beschr¨ankte finite-state-Generizit¨at und Spezialf ¨alle hiervon, die wir durch die Beschr¨ankung auf totale Erweiterungsfunktionen bzw. auf Erweiterungen konstanter L¨ange erhalten. Wir geben eine rein kombinatorische Charakterisierung der beschr¨ankt finite-state-generischen Mengen: Diese sind gerade die Mengen, deren charakteristische Folge saturiert ist, d.h. jedes Bin¨arwort als Teilwort enth¨alt. Mit Hilfe dieser Charakterisierung bestimmen wir die Komplexit¨at der beschr¨ankt finitestate- generischen Mengen und zeigen, dass solch eine generische Menge nicht regul¨ar sein kann es aber kontext-freie Sprachen mit dieser Generizit¨atseigenschaft gibt. Die von uns betrachteten unbeschr¨ankten finite-state-Generizit¨atskonzepte basieren auf Moore-Funktionen und auf Verallgemeinerungen dieser Funktionen. Auch hier vergleichen wir die St¨arke der verschiedenen korrespondierenden Generizit¨atskonzepte und er¨ortern die Frage, inwieweit diese Konzepte m¨achtiger als die beschr¨ankte finite-state-Generizit ¨at sind. Unsere Untersuchungen der finite-state-Generizit¨at beruhen zum Teil auf neuen Ergebnissen ¨uber Bi-Immunit¨at in der Chomsky-Hierarchie, einer neuen Chomsky-Hierarchie f¨ur unendliche Folgen und einer gr¨undlichen Untersuchung der saturierten Folgen. Diese Ergebnisse – die von unabh¨angigem Interesse sind – werden im ersten Teil der Dissertation vorgestellt. Sie k¨onnen unabh¨angig von dem Hauptteil der Arbeit gelesen werden

Automated test generation from algebraic specifications

PhD ThesisThis thesis is a contribution to work on the specification-based testing of computing systems. The development of computing systems is a challenging task. A great deal of research has been directed at support for analysis, design and implementation aspects, yielding a wide range of development techniques. However, the crucial area of system testing remains relatively under-explored. Because a project may spend a good part of its budget on testing, even modest improvements to the cost-effectiveness of testing represent substantial improvements in project budgets. Relatively little literature has been devoted to the entire testing process, including specification, generation, execution and validation. Most of the academic literature seems to assume a revolutionary change of the testing framework. On the contrary industry follows a more traditional approach consisting of trusted methods and based on personal experience. There is a need for testing methods that improve the effectiveness of testing but do so at reasonable cost and which do not require a revolutionary change in the development technology. The novel goal of the work described in this thesis is to "lift" traditional testing so that it takes advantage of system specifications. We provide a framework - hepTEsT- which is motivated by this goal. To that end, hepTEsT is a framework consisting of a specification language, a technology for generating tests in accordance with test strategies, a means of applying the tests to the implementations and support for validation of outcomes against the specification-based tests. We will first categorise different testing methodologies and then examine some of the past and present approaches to test data: we develop only the necessary theoretical foundations for hepSPEc and always consider the requirements of testing. The formalism hepSPEc for system description is based upon a well-defined algebraic approach. It utilises a novel approach allowing the description of finite domains in a way suitable for engineering purposes. The engineers' tasks are to provide an adequate description of the system in hepSPEC. The approach proposed in this thesis is grounded in the traditional approach to testing where test data is provided to the system under test and the outcome is compared to the expected outcome. To enhance the capabilities of the framework a general order on test inputs is proposed to be used in test strategies. Traditional testing strategies requiring an order on test inputs are introduced and their realisation in hepTEsT discussed as well as a proposal of new strategies which lend themselves to this particular approach. The manipulation of the specification yields abstract test cases which are then transformed into test cases suitable for the chosen implementation of the system. This transformation, called test reification, is necessary to bridge the "abstraction gap" between the abstract specification-derived tests and the concrete implementation on which the test must run. The transformation is necessary in order for the approach to be practical and is achieved through homomorphisms which are expressed in specially adapted grammars. This transformation is also applied to the generated test outcome and is aimed there at easing test result validation. The utility of the hepTEsT approach is illustrated by means of a simple example, a larger case study and one carried out within the aviation industry