Learning algebraic structures from text

AbstractThe present work investigates the learnability of classes of substructures of some algebraic structures: submonoids and subgroups of given groups, ideals of given commutative rings, subfields of given vector spaces. The learner sees all positive data but no negative one and converges to a program enumerating or computing the set to be learned. Besides semantical (BC) and syntactical (Ex) convergence also the more restrictive ordinal bounds on the number of mind changes are considered. The following is shown: (a) Learnability depends much on the amount of semantic knowledge given at the synthesis of the learner where this knowledge is represented by programs for the algebraic operations, codes for prominent elements of the algebraic structure (like 0 and 1 fields) and certain parameters (like the dimension of finite-dimensional vector spaces). For several natural examples, good knowledge of the semantics may enable to keep ordinal mind change bounds while restricted knowledge may either allow only BC-convergence or even not permit learnability at all.(b) The class of all ideals of a recursive ring is BC-learnable iff the ring is Noetherian. Furthermore, one has either only a BC-learner outputting enumerable indices or one can already get an Ex-learner converging to decision procedures and respecting an ordinal bound on the number of mind changes. The ring is Artinian iff the ideals can be Ex-learned with a constant bound on the number of mind changes, this constant is the length of the ring. Ex-learnability depends not only on the ring but also on the representation of the ring. Polynomial rings over the field of rationals with n variables have exactly the ordinal mind change bound ωn in the standard representation. Similar results can be established for unars. Noetherian unars with one function can be learned with an ordinal mind change bound aω for some a

Performance modelling and the representation of large scale distributed system functions

This thesis presents a resource based approach to model generation for performance characterization and correctness checking of large scale telecommunications networks. A notion called the timed automaton is proposed and then developed to encapsulate behaviours of networking equipment, system control policies and non-deterministic user behaviours. The states of pooled network resources and the behaviours of resource consumers are represented as continually varying geometric patterns; these patterns form part of the data operated upon by the timed automata. Such a representation technique allows for great flexibility regarding the level of abstraction that can be chosen in the modelling of telecommunications systems. None the less, the notion of system functions is proposed to serve as a constraining framework for specifying bounded behaviours and features of telecommunications systems. Operational concepts are developed for the timed automata; these concepts are based on limit preserving relations. Relations over system states represent the evolution of system properties observable at various locations within the network under study. The declarative nature of such permutative state relations provides a direct framework for generating highly expressive models suitable for carrying out optimization experiments. The usefulness of the developed procedure is demonstrated by tackling a large scale case study, in particular the problem of congestion avoidance in networks; it is shown that there can be global coupling among local behaviours within a telecommunications network. The uncovering of such a phenomenon through a function oriented simulation is a contribution to the area of network modelling. The direct and faithful way of deriving performance metrics for loss in networks from resource utilization patterns is also a new contribution to the work area

Philosophical logics - a survey and a bibliography

Intensional logics attract the attention of researchers from differing academic backgrounds and various scientific interests. My aim is to sketch the philosophical background of alethic, doxastic, and deontic logics, their formal and metaphysical presumptions and their various problems and paradoxes, without attempting formal rigor. A bibliography, concise on philosophical writings, is meant to allow the reader\u27s access to the maze of literature in the field

The use of data-mining for the automatic formation of tactics

This paper discusses the usse of data-mining for the automatic formation of tactics. It was presented at the Workshop on Computer-Supported Mathematical Theory Development held at IJCAR in 2004. The aim of this project is to evaluate the applicability of data-mining techniques to the automatic formation of tactics from large corpuses of proofs. We data-mine information from large proof corpuses to find commonly occurring patterns. These patterns are then evolved into tactics using genetic programming techniques

Semantic spaces

Any natural language can be considered as a tool for producing large databases (consisting of texts, written, or discursive). This tool for its description in turn requires other large databases (dictionaries, grammars etc.). Nowadays, the notion of database is associated with computer processing and computer memory. However, a natural language resides also in human brains and functions in human communication, from interpersonal to intergenerational one. We discuss in this survey/research paper mathematical, in particular geometric, constructions, which help to bridge these two worlds. In particular, in this paper we consider the Vector Space Model of semantics based on frequency matrices, as used in Natural Language Processing. We investigate underlying geometries, formulated in terms of Grassmannians, projective spaces, and flag varieties. We formulate the relation between vector space models and semantic spaces based on semic axes in terms of projectability of subvarieties in Grassmannians and projective spaces. We interpret Latent Semantics as a geometric flow on Grassmannians. We also discuss how to formulate G\"ardenfors' notion of "meeting of minds" in our geometric setting.Comment: 32 pages, TeX, 1 eps figur

Property Theories

Revised and reprinted; originally in Dov Gabbay & Franz Guenthner (eds.), Handbook of Philosophical Logic, Volume IV. Kluwer 133-251. -- Two sorts of property theory are distinguished, those dealing with intensional contexts property abstracts (infinitive and gerundive phrases) and proposition abstracts (‘that’-clauses) and those dealing with predication (or instantiation) relations. The first is deemed to be epistemologically more primary, for “the argument from intensional logic” is perhaps the best argument for the existence of properties. This argument is presented in the course of discussing generality, quantifying-in, learnability, referential semantics, nominalism, conceptualism, realism, type-freedom, the first-order/higher-order controversy, names, indexicals, descriptions, Mates’ puzzle, and the paradox of analysis. Two first-order intensional logics are then formulated. Finally, fixed-point type-free theories of predication are discussed, especially their relation to the question whether properties may be identified with propositional functions

A framework for understanding what algebraic thinking is

In relation to the learning of mathematics, algebra occupies a very special place, both because it is in itself a powerful tool for solving problems and modelling situations, and also because it is essential to the learning of so many other parts of mathematics. On the other hand, the teaching of algebra has proven to be a difficult task to accomplish, to the extent of algebra itself being sometimes considered the border line which separates those who can from those who cannot learn mathematics. A review of the research literature shows that no clear characterisation of the algebraic activity has been available, and that for this reason research has produced only a local understanding of aspects of the learning of algebra. The research problem investigated in this dissertation is precisely to provide a clear characterisation of the algebraic activity. Our research has three parts: (i) a theoretical characterisation of algebraic thinking, which is shown to be distinct from algebra; in our framework we propose that algebraic thinking IS • thinking aritmnetically, • thinking internally, and • thinking analytically. and each of those characteristics are explained and analysed; (ii) a study of the historical development of algebra and of algebraic thinking; in this study it is shown that our characterisation of algebraic thinking provides an adequate framework for understanding the tensions involved in the production of an algebraic knowledge in different historically situated mathematical cultures, and also that the characteristics of the algebraic knowledge of each of those mathematical cultures can only be understood in the context of their broader assumptions, particularly in relation to the concept of number. (iii) an experimental study, in which we examine the models used by secondary school students, both from Brazil and from England, to solve "algebraic verbal problems" and "secret number problems"; it is shown that our characterisation of algebraic thinking provides an adequate framework for distinguishing different types of solutions, as well as for identifying the sources of errors and difficulties in those students' solutions. The key notions elicited by our research are those of: (a) intrasystemic and extrasystemic meaning; (b) different modes of thinking as operating within different Semantical Fields; (c) the development of an algebraic mode of thinking as a process of cultural immersion- both in history and for individual learners; (d) ontological and symbolical conceptions of number, and their relationship to algebraic thinking and other modes of manipulating arithmetical relationships; (e) the arithmetical articulation as a central aspect of algebraic thinking; and, (f) the place and role of algebraic notation in relation to algebraic thinking. The findings of our research show that although it can facilitate the learning of certain early aspects of algebra, the use of non-algebraic models-such as the scale balance or areas-to "explain" particular algebraic facts, contribute, in fact, to the constitution of obstacles to the development of an algebraic mode of thinking; not only because the sources of meaning in those models are completely distinct from those in algebraic thinking, but also because the direct manipulation of numbers as measures, by manipulating the objects measured by the numbers, is deeply conflicting with a symbolic understanding of number, which is a necessary aspect of algebraic thinking